Plainsound Harmonic Space Calculator | basic User Guide Thomas Nicholson & Marc Sabat send bug reports to [email protected] Description The Plainsound Harmonic Space Calculator | basic (abbreviated as HS CALC | basic) is a tool for composers and musicians who are interested in discovering and working with properties of intervals tuned in just intonation. One of HS CALC | basic’s key objectives is to supplement and encourage use of the Extended Helmholtz-Ellis JI Pitch Notation (HEJI) developed by Marc Sabat and Wolfgang von Schweinitz. HEJI is based on the Pythagorean series of untem- pered perfect fifths, notated on the traditional five-line staff using flats, naturals, and sharps. It explicitly notates the raising and lowering of these pitches by spe- cific microtonal ratios, defined for each prime, providing visually distinctive logos distinguishing families of justly tuned intervals that relate to the harmonic series. These take the form of strings of additional accidental symbols based on historical precedents. HS CALC | basic allows users to compute and compare relative pitch heights of any frequencies based on a chosen reference. Pitches may be inputted in HEJI notation and/or as ratios; outputs include HEJI, ratios, absolute cents, tuning meter read- out, software pitch bends, steps in an arbitrary equal division of the octave, etc. Version 3.0 (2020-06-26) Contents 1 Basic Workflow ................................... 2 1.1 System Requirements . 2 2 The REFERENCE Area ............................. 2 2.1 Tuning meter setting . 3 2.2 EDO cross-comparison . 3 3 The INPUT Areas ................................. 4 3.1 HEJI notation palette . 4 3.2 JI ratio . 4 3.3 Moving between input methods . 5 4 The OUTPUT Area ................................ 5 5 The MELODIC DISTANCE Area ....................... 7 User Guide : Plainsound Harmonic Space Calculator | basic Page 2 1 Basic Workflow In general, movement of information through the calculator flows from left to right across the screen on desktop and top to bottom on mobile. After setting the REFERENCE (1/1), just intonation relationships—whether as notation or ratios—may be entered into the calculator using the HEJI notation palette or the JI ratio input fields (the desired input method must be selected). The calculator computes various types of information about the INPUT, including no- tation, ratio, tuning meter cent deviation, frequency, software pitch bend values, etc. The results are displayed in the OUTPUT area. Precision (i.e. number of decimal places) may be specified with the calculator precision drop-down menu. Additionally, two pitches may be compared to determine their MELODIC DISTANCE (i.e. the interval between them). Ratios may be directly entered in the input fields or “loaded” by re-routing previously calculated pitches from the OUTPUT area. 1.1 System Requirements HS CALC | basic is a simple JavaScript application able to run in any HTML5-compatible web browser on a wide variety of system configurations. On desktop, the app has been optimised for full HD and higher resolution displays (i.e. 1080p or greater). On displays with lower resolutions (e.g. 720p HD), “areas” may reflow to form new rows. On mobile, the areas form a single vertical column. The calculator requires an active connection to the internet in order to guarantee proper functionality. For information about offline compatibility, please contact Thomas Nicholson at [email protected]. 2 The REFERENCE Area By default, HS CALC | basic’s harmonic reference (1/1) is the pitch written as jA in SPN octave 4 and tuned to 440 Hz. This may be considered a fundamental or a generator for all the other tones in the harmonic space, similar to Harry Partch’s concept of Monophony. In Genesis of a Music, he describes it as “an organization of musical materials based upon the faculty of the human ear to perceive all intervals and to deduce all principles of musical relationship as an expansion from unity, as 1 is to 1.” Changing the REFERENCE pitch’s octave, diatonic pitch, and/or accidental automati- cally updates its frequency—1/1 frequency (Hz)—as a 12-EDO tempered pitch with respect to A4 440 Hz. This frequency may subsequently be adjusted to any desired value. Example. Selecting octave 4, tempered diatonic pitch G, and tempered accidental j automatically calculates a frequency of 391.9954 Hz, a 12-EDO wholetone below A4 440 Hz (440 ÷ 21/6). To work in the harmonic space of Harry Partch, the user may adjust this value to 392.0000 Hz. User Guide : Plainsound Harmonic Space Calculator | basic Page 3 2.1 Tuning meter setting The options under tuning meter setting affect how outputted pitches are read on a tuning meter by controlling the frequency of the tuning meter’s own reference or “Kammerton” (typically tempered A4 set to 440 Hz). The default setting, 1/1 = 0 cents, automatically computes the required frequency of tempered A4 in the meter’s settings such that the selected reference pitch 1/1 will read ±0 cents. Example. If 1/1 is defined to be jG4 at 392 Hz, then nA4 (9/8) is at 441 Hz (a step of 203.91 cents from jG4); the expected tuning meter indication would read A+3:91 cents. However, if the frequency of tempered jA4 on the tuning meter (Kammerton) were left at 440 Hz, then the required cent deviation of nA4 is slightly larger than the expected display value (A+3:93 cents in order to account for the redefinition of tempered jG4 as 392 Hz). The calculator automatically updates the required frequency of tempered jA4 (i.e. jA4 on meter is 440.0051 Hz). Toggling 1/1 cents relative to meter setting allows for independent control over the desired frequency of 1/1 as well as the relative cent values indicated by a tuning meter. Adjusting the frequency of tempered A4 in the meter’s settings allows any pitch to read ±0 cents. Example. Sometimes it may be useful to redefine 1/1—e.g. the note jG4 at 392 Hz—but allow some other pitch to remain at 0 cents—e.g. nA4 (9/8). With 1/1 cents relative to meter setting selected, calculated cent devia- tion output may be arbitrarily redefined by directly adjusting the frequency of tempered A4 on meter. If the frequency is changed to 441 Hz, then the fre- quency of Pythagorean A4 (9/8) will be equal to the frequency of tempered ± n − jA4, reading A 0 cents, while 1/1 will read G 3:91 cents. Example. Catherine Lamb has written a series of pieces based on a 10 Hz − fundamental, choosing to notate this 1/1 as eE 1. Since each partial is a multiple of 10, the Kammerton frequency on the tuning meter may be made equal to the 44th partial (written 4eA4 sounding with frequency 440 Hz) in order to provide practical cent indications for performers. This allows for a potential simplification of the notation, where all of the pitches may be written a quartertone lower than sounding (with respect to the tuning meter)—i.e. the fundamental is notated as the pitch-class eE but sounds as eE+48:7 cents. 2.2 EDO cross-comparison A reference equal temperament (EDO, “equal divisions of the octave”) may be specified as a cross-comparison to untempered just intonation. Outputs are mapped to steps in the specified EDO (default 53-EDO). For more information, reference section 4 (“ratio in EDO steps + error”). User Guide : Plainsound Harmonic Space Calculator | basic Page 4 3 The INPUT Areas Before inputting pitches into HS CALC | basic, the desired input method must be selected. • HEJI notation palette (default) • JI ratio (input × offset) 3.1 HEJI notation palette The HEJI notation palette is straightforward: an input pitch may be defined according to its SPN octave, Pythagorean diatonic pitch, and HEJI accidental(s). Palette input accidentals are limited to allow up to three Pythagorean apotomes (±) and three steps (±) in each of the prime dimension 5 through 31 (see below for the ratios used in the HEJI accidentals). Prime dimension HEJI symbol Symbol alters by 5 dmu | fow 81/80 7 < | > 64/63 11 5 | 4 33/32 13 0 | 9 27/26 17 : | ; 2187/2176 19 * | / 513/512 23 6 | 3 736/729 29 7 | 2 261/256 31 1 | 8 32/31 37 à | á 37/36 41 - | + 82/81 43 è | é 129/128 47 ì | í 752/729 3.2 JI ratio Ratio input comprises two components. • the offset ratio – a local 1/1 (with respect to the global 1/1 defined in REFERENCE) • the input ratio – a ratio which is automatically multiplied by the offset ratio Example. If the offset ratio is 1/1 and the input ratio is 10/9, then the OUTPUT ratio is calculated as 10/9 with respect to the REFERENCE. However, if the offset ratio were set to 9/8, then the OUTPUT ratio would be calculated as 10/9 with respect to the REFERENCE offset by 9/8 (the offset ratio 9/8 functions as a local 1/1). The result is a compounding of the two input ratios, outputting 5/4 (10/9 × 9/8) with respect to the REFERENCE. Whole numbers or decimal values may be used (the calculator will automatically convert decimals into whole number fractions when possible). User Guide : Plainsound Harmonic Space Calculator | basic Page 5 3.3 Moving between input methods Making changes in an inactive input method will have no effect on the output until that method is made active (i.e. selected). However, calculated outputs (irrespective of the active input method) may be loaded at any time into the offset ratio and the input ratio fields by means of the load output buttons.